Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching

Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\mathrm{\ell }=\lfloor \frac{n}{2}\rfloor$. We prove the sharp pointwise estimate $$q_2(E)\ge -\frac{2(\mathrm{\ell }-1)}{3\mathrm{\ell }}\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}(E){\mathrm{I}\mathrm{d}}_{{\mathrm{\Lambda }}^2{V}^{\ast }}$$ for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition ${\mathrm{Rm}}_g=K_{\min }I+E$, we obtain the vanishing of $H^2(M;\mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields $b_2(M)=0$ in odd dimensions and $b_2(M)\le 1$ in even dimensions. At even-dimensional endpoint, $b_2(M)>0$ forces $(M,g)$ to be isometric, up to scaling, to ${\mathbb{C}\mathbb{P}}^{\mathrm{\ell }}$ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion $$K_{\min }\ge \frac{n(n-1)}{n^2-n+12}{S}_0⟹\mathrm{P}\mathrm{I}\mathrm{C}2\mathrm{.}$$ The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.